B. P˚uˇza and Z. Sokhadze
ON THE WEIGHTED INITIAL PROBLEM FOR SINGULAR FUNCTIONAL DIFFERENTIAL SYSTEMS
Abstract. For singular functional differential systems, sufficient condi- tions for solvability and well-posedness of the weighted initial problem are established.
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2010 Mathematics Subject Classification. 34A12, 34K05, 34K10.
Key words and phrases. Singular functional differential system, the weighted initial problem, solvability, well-posedness.
In a finite interval ]a, b[ we consider the functional differential system dx(t)
dt =f(x)(t) (1)
with the weighted initial condition lim sup
t→a
°°φ−1(t)x(t)°
°<+∞. (2)
Here, f : C([a, b];Rn) → Lloc(]a, b];Rn) is a singular operator satisfying the local Carath´eorory conditions,φ(t) = diag¡
ϕ1(t), . . . , ϕn(t)¢
, and ϕi : [a, b]→R+(i= 1, . . . , n) are continuous non-decreasing functions such that ϕi(a) = 0, ϕi(t)>0 fora < t≤b(i= 1, . . . , n).
The initial problem for the singular system (1) has been thoroughly in- vestigated in the cases, in whichf is either the Nemytski’s operator [1]–[6], or the evolutionary operator [7]–[9]. The weighted initial problem for higher order singular functional differential equations is studied in [11]–[14]. As for the weighted singular problem (1), (2), it is not studied well enough. In the present paper unimprovable in a certain sense conditions are given which, respectively, guarantee solvability and well-posedness of this problem.
Throughout the paper, the use will be made of the following notation.
R= ]− ∞,+∞[ ,R+= [0,+∞[ .
Rn is the space of n-dimensional real column-vectors x= (xi)ni=1 with the norm
kxk= Xn i=1
|xi|.
Ifx= (xi)ni=1∈Rn, then [x]+=
³xi+|xi| 2
´n
i=1.
r(X) is the spectral radius of then×nmatrixX, andX−1is the inverse toX matrix.
diag(x1, . . . , xn) is the diagonal n×n-matrix with diagonal elements x1, . . . , xn.
IfX = diag(x1, . . . , xn), then Sgn(X) =¡
sgn(x1), . . . ,sgn(xn)¢ .
Rn+ and Rn×n+ are the sets ofn-dimensional vectors and n×n-matrices with nonnegative components.
C([a, b];Rn) is the space of continuous vector functions x : [a, b]→ Rn with the norm
kxkC= max n
kx(t)k: a≤t≤b o
.
Cφ([a, b];Rn) is the space of continuous vector functionsx: [a, b]→Rn, satisfying the condition (2), with the norm
kxkCφ = supn°°φ−1(t)x(t)°
°: a < t≤bo . Ifx= (xi)ni=1∈Cφ([a, b];Rn), then
|x|Cφ =¡ kxikCϕi
¢n
i=1.
L([a, b];Rn) is the space of vector functions with Lebesgue integrable on [a, b] components.
Lloc(]a, b];Rn) is the space of vector functions whose components are Lebesgue integrable on [a+ε, b] for an arbitrarily smallε >0.
Kloc(]a, b]×Rk;Rm) and Kloc(C([a, b];Rk);Lloc(]a, b];Rm)) are the sets of vector functions g : ]a, b]×Rk → Rm and operators f : C([a, b];Rk)→ Lloc(]a, b];Rm), satisfying the local Carath´eodory conditions (see [15]).
An important particular case of the functional differential system (1) is the differential system with a deviating argument
dx(t) dt =g¡
t, x(t), x(τ(t))¢
. (3)
Along with the problem (1), (2), we consider the problem (3), (2). Every- where below, when the question concerns these problems, it will be assumed that
f ∈Kloc
¡C([a, b];Rn);Lloc(]a, b];Rn)¢
, g∈Kloc(]a, b]×R2n;Rn), andτ: [a, b]→[a, b] is a measurable function.
We are mainly interested in the case, where the systems (1) and (3) are singular, i.e., in the case in which
Zb a
fρ∗(t)dt= +∞ and Zb a
gρ∗(t)dt= +∞ for ρ >0,
where
fρ∗(t) = supn°
°f(x)(t)°
°: kxkC≤ρ o
, gρ∗(t) = maxn°
°g(t, x, y)°
°: kxk+kyk ≤ρo . For an arbitrary positive numberδ, we put
χ(t, δ, λ) =
(0 for a≤t < a+δ λ for t > a+δ , and consider the auxiliary initial problem
dx(t)
dt =χ(t, δ, λ)f(x)(t), (4)
x(a) = 0, (5)
depending on the parametersλ∈]0,1] andδ >0.
On the basis of Corollary 2 in [16], the following theorem can be proved.
Theorem 1. Let there exist a positive numberρ0 such that for arbitrary λ ∈]0,1] and δ > 0 every solution x of the problem (4),(5) admits the estimate
kxkCφ ≤ρ0.
Then the problem (1),(2) has at least one solution.
This theorem allows one to get efficient sufficient conditions for the solv- ability of the problems (1), (2) and (3), (2). In particular, the following propositions are valid.
Theorem 2. Let there exist a matrixP ∈Rn×n+ and a vector function q:R+ →Rn+ such that
r(P)<1, lim
ρ→+∞
kq(ρ)k
ρ = 0, (6)
and for an arbitrary vector functionx∈Cφ([a, b];Rn)on the interval [a, b]
the inequality Zt a
h
sgn(x(s))f(x)(s) i
+ds≤φ(t)
³
P|x|Cφ+q¡ kxkCφ
¢´
is fulfilled. Then the problem (1),(2)has at least one solution.
Corollary 1. Let the functionsϕi(i= 1, . . . , n)be absolutely continuous and let there exist a set of zero measureI0⊂[a, b], matricesPk∈Rn×n+ (k= 1,2) and a vector function q : R+ → Rn+ with non-decreasing components such that on the set([a, b]\I0)×R2n the inequality
Sgn(x)g(t, x, y)≤φ0(t)
³
P1φ−1(t)|x|+P2φ−1(τ(t))|y|
´ + +φ0(t)q³°
°φ−1(t)|x|+φ−1(τ(t))|y|°
°´
is fulfilled. If, moreover, the conditions (6)are fulfilled, whereP=P1+P2, then the problem (3),(2) has at least one solution.
Remark 1. In Theorem 2 and Corollary 1, the condition r(P) < 1 is unimprovable and it cannot be replaced by the condition r(P) ≤ 1. The validity of that fact follows directly from the theorem below.
Theorem 3. Let the functionsϕi (i= 1, . . . , n)be absolutely continuous and let there exist a set of zero measure I0 ⊂ [a, b], matrices Pk ∈ Rn×n+ (k = 1,2) and a vector q0 = (q0i)ni=1 with positive components q0i (i = 1, . . . , n)such that on the set([a, b]\I0)×R2n the inequality
g(t, x, y)≥φ0(t)
³
P1φ−1(t)|x|+P2φ−1(τ(t))|y|+q0
´
is fulfilled. If, moreover, r(P1+P2)≥1, then the problem (3),(2) has no solution.
Along with the problem (1), (2), we consider the perturbed problem dy(t)
dt =f(y)(t) +h(t), (7)
lim sup
t→a
°°φ−1(t)y(t)°
°<+∞, (8)
and introduce the following
Definition. The problem (1), (2) is called well-posed if there exists a positive number ρ such that for an arbitrary function h ∈ L([a, b];Rn), satisfying the condition
νφ(h) = sup
½°°°φ−1(t) Zt a
|h(s)|ds
°°
°: a < t≤b
¾
<+∞,
the problem (7), (8) is uniquely solvable and its solution admits the estimate ky−xkCφ ≤ρνφ(h),
wherexis a solution of the problem (1), (2).
Theorem 4. Let there exist a matrixP ∈Rn×n+ such thatr(P)<1, and for arbitrary vector functionsx andy ∈Cφ([a, b];Rn)in the interval [a, b]
the inequality Zt a
h
sgn(y(s))¡
f(x+y)(s)−f(x)(s)¢i
+ds≤φ(t)P|y|Cφ
is fulfilled. If, moreover,
sup
½°°°φ−1(t) Zt a
¯¯f(0)(s)¯
¯ds
°°
°: a < t≤b
¾
<+∞, then the problem (1),(2) is well-posed.
Corollary 2. Let the functionsϕi(i= 1, . . . , n)be absolutely continuous and let there exist a set of zero measureI0⊂[a, b]and matricesPk∈Rn×n+ (k= 1,2)such that r(P1+P2)<1, and for anyt∈[a, b]\I0,x,x,y and y∈Rn the inequality
sgn(x)³ g¡
t, x+x, y+y¢
−g(t, x, y)´
≤φ0(t)³
P1φ−1(t)|x|+P2φ−1(τ(t))|y|´ is fulfilled. If, moreover,
sup
½°°°φ−1(t) Zt a
¯¯g(s,0,0)¯¯ds
°°
°: a < t≤b
¾
<+∞, then the problem (3),(2) is well-posed.
From Theorem 3 and Corollary 2 it follows
Corollary 3. Let the functionsϕi(i= 1, . . . , n)be absolutely continuous and
g(t, x, y) =φ0(t)
³
P1φ−1(t)|x|+P2φ−1(τ(t))|y|+q0
´ ,
where Pk ∈ Rn×n+ (k = 1,2), and q0 ∈ Rn+ is the vector with positive components. Then the problem (3),(2)is well-posed if and only if
r(P1+P2)<1.
Remark 2. According to Corollary 3, the inequality r(P)< 1 (r(P1+ P2)<1) in Theorem 4 (in Corollary 2) is unimprovable and it cannot be replaced by the inequalityr(P)≤1 (r(P1+P2)≤1).
Acknowledgement
For the first author this work is supported by RVO:67985840, and for the second author – by the Shota Rustaveli National Science Foundation (Project # GNSF/ST09−175−3-101).
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(Received 24.09.2012) Author’s address:
B. P˚uˇza
Mathematical Institute, Academy of Sciences of the Czech Republic, branch in Brno, ˇZiˇzkova 22, 616 62 Brno, Czech Republic.
E-mail: [email protected] Z. Sokhadze
Akaki Tsereteli State University, 59, Queen Tamar St., Kutaisi 4600, Georgia.
E-mail: [email protected]
